1. Karlheinz Schwarz Institute of Materials Chemistry TU Wien
Magnetism (FM, AFM, FSM)
Karlheinz Schwarz
Institute of Materials Chemistry
TU Wien

2. Localized vs. itinerant systems
In localized systems (e.g. some rare earth) the magnetism is mainly governed by the atom (Hund‘s rule)
In itinerant (delocalized) systems (many transition metals) magnetism comes from partial occupation of states, which differ between spin-up and spin-down.
Boarderline cases (some f-electron systems)
details of the structure (e.g. lattice spacing) determine whether or not some electrons are localized or itinerant.

3. Ferro-, antiferro-, or ferri-magnetic
Ferromagnetic (FM) (e.g. bcc Fe)
M > 0
Antiferromagnetic (AFM) (e.g. Cr)
M = 0
Ferrimagnetic cases
the moments at different atoms are antiparallel but of different magnitude
Non-collinear magnetism (NCM)
the magnetic moments are not ligned up parallel.

4. Itinerant electron magnetism
Experimental facts:
Curie
temperature

5. Stoner theory of itinerant electron magnetism
The carriers of magnetism are the unsaturated spins
in the d-band.
Effects of exchange are treated with a
molecular field term.
3. One must conform to Fermi statistics.
Stoner, 1936

6. Stoner model for itinerant electrons
In a
non magnetic (NM) case
N↑ = N↓ (spin-up and spin-down)
ferromagnetic (FM) case
N↑ > N↓ (majority and minority spin)
the moments at all sites are parallel
(collinear)
the (spin) magnetic moment m
m = N↑ - N↓
its orientation with respect to the crystal axes is only defined by
spin orbit coupling.
there can also be an orbital moment
it is often suppressed in 3d transition metals
Exchange splitting
spin-down spin-up
exchange interaction
Stoner criterion

7. Stoner model for itinerant electrons
The existence of ferromagnetism
(FM) is governed by the
Stoner criterion
I . N(EF) > 1
N(EF) DOS at EF (of NM case)
I Stoner parameter
~ independent of structure
Ferromagnetism appears when the gain in exchange energy is
larger than the loss in kinetic energy
(a) Fe 1
IFe
(b) Ni 1
INi
P.James, O.Eriksson, B.Johansson, I.A.Abrikosov,
Phys.Rev.B 58, ... (1998)

8. bcc Fe ferromagnetic case Exchange splitting EF at high DOS
Non magnetic case
spin-up
spin-down
spin-up
spin-down EF EF
Exchange splitting
EF at high DOS

9. DFT ground state of iron
LSDA NM
fcc
in contrast to
experiment
GGA FM
bcc
Correct lattice constant
Experiment
LSDA
GGA
GGA
LSDA

10. Iron and its alloys Fe: weak ferromagnet (almost)
Co: strong ferromagnet

11. Magnetism and crystal structure
V. Heine: „metals are systems with
unsaturated covalent bonds“

12. Covalent magnetism Fe-Co alloys
e.g. Fe-Co alloys
Wigner delay times

13. Spin projected DOS of Fe-Co alloys
The alloy is represented by ordered structures
Fe3Co and FeCo3 (Heusler)
FeCo Zintl or CsCl
Fe, Co bcc

14. Iron and its alloys
Itinerant or localized?

15. Magnetization density in FeCo
Magnetization density difference between
Majoity spin
Minority spin
Localized around
Fe and Co
slightly negative between the atoms
Itinerant electrons
m(r)=ρ (r)-ρ (r)
CsCl
structure
K.Schwarz, P.Mohn, P.Blaha, J.Kübler,
Electronic and magnetic structure of bcc Fe-Co alloys from band theory,
J.Phys.F:Met.Phys. 14, 2659 (1984)

16. Bonding by Wigner delay time
single scatterer (Friedel)
Bessel
Neumann
V(r)=0 solution:
Rl joined in value and
slope defines phase shift :
Friedel sum
Wigner delay time

17. Phase shifts, Wigner delay times of Fe, Co, Ni
resonance states
Wigner
delay time
Phase
shifts

18. Covalent magnetism in FeCo
Wigner delay time
For spin up
Fe and Co equivalent
partial DOS similar
typical bcc DOS
For spin down
Fe higher than Co
antibonding Co Fe Co
bonding
No charge transfer
between Fe and Co

19. Magnetism and crystal structure
Covalent magnetism, FeCo:

20. Antiferromagnetic (AFM) Cr
Cr has AFM bcc structure
There is a symmetry
it is enough to do the spin-up
calculation
spin-down can be copied
Cr1
Cr2
Cr1
spin-up
spin-up
Cr2 EF
spin-down
spin-down
Cr1 = Cr2
Cr2 = Cr1

21. Zeolite, sodalite Si Al Al-silicate corner shared  cage
SiO4 tetrahedra
AlO4 tetrahedra
 cage
Al / Si ratio 1
alternating
ordered (cubic)
3 e- per cage Si Al

22. SES Sodium electro sodalite
Si-Al zeolite (sodalite)
Formed by corner-shared SiO4 and AlO4 tetrahedra
Charge compensated by
doping with
4 Na+
one e- (color center)
antiferromagnetic (AFM) order
of e-
color center e-
Energy (relative stability)

23.   SES AFM order between color centers (e-) Spin density  - 
G.K.H. Madsen, Bo B. Iversen, P. Blaha, K. Schwarz,
Phys. Rev. B 64, (2001)

24. INVAR alloys (invariant)
e.g. Fe-Ni
Such systems essentially show
no thermal expansion
around room temperature

25. INVAR (invariant) of Fe-Ni alloys
The thermal expansion of the Eifel tower
Measured with a rigid Fe-Ni INVAR wire
The length of the tower correlates with the temperature
Fe65Ni35 alloy has vanishing thermal expansion around room temperature
Ch.E.Guillaume (1897)

26. Magnetostriction and Invar behaviour
What is magnetostriction?
Magnetostriction ws0 is the
difference in volume between the
volume in the magnetic ground
state and the volume in a
hypothetical non-magnetic state.
Above the Curie temperature the
magnetic contribution wm vanishes. Tc

27. Fe-Ni Invar alloys „classical“ Fe-Ni Invar
Fe65Ni35 alloy has vanishing thermal expansion around room temperature

28. Early explanations of INVAR
R.J.Weiss
Proc.Roy.Phys.Soc (London) 32, 281 (1963)
FCC
fcc Fe
50% Fe
low spin
m=0.5 μB AF
a = 3.57 Å
high spin
m=2.8 μB FM
a = 3.64 Å
75% Fe
small moment
γ1 AF
small volume kT
100% Fe
high moment
γ2 FM
large volume
volume (Bohr)3
A.R.Williams, V.L.Moruzzi, G.D.Gelatt Jr., J.Kübler, K.Schwarz,
Aspects of transition metal magnetism,
J.Appl.Phys. 53, 2019 (1982)

29. Energy surfaces of Fe-Ni alloys
This fcc structure
from non magnetic Fe (fcc)
to ferromagnetic Ni
as the composition changes
At the INVAR composition
There is a flat energy surface
as function of volume and moment
% Fe
100%
75%
50% 0%

30. Finite temperature T Energy surface at T=0 (DFT) Finite temperature
as a function of volume and moment
using fixed spin moment (FSM) calculations
Finite temperature
Spin and volume fluctuations
Ginzburg-Landau model T
439 K TC
300 K
200 K
100 K
0 K

31. allows to explore energy surface E(V,M) as function of
FSM calculations
fixed spin moment (FSM)
e.g. Fe-Ni alloy
allows to explore energy surface E(V,M)
as function of
volume V
magnetic moment M

32. Fixed spin moment (FSM) method
There are systems (e.g. like fcc Fe or fcc Co), for which the magnetization shows a hysteresis, when a magnetic field is applied (at a volume V).
The volume of the unit cell defines the Wigner-Seitz radius rWS
The hysteresis causes numerical difficulties, since there are several solutions (in the present case 3 for a certain field H).
In order to solve this problem the FSM method was invented
Hysteresis

33. Fixed spin moment (FSM) method
Conventional scheme constrained (FSM) method
output
output
input
Simple case:
bcc Fe
one SCF
many
calculations
difficult case:
Fe3Ni
poor convergence
good convergence

34. FSM Physical situation: One applies a field H and obtains M
but this functions can be multivalued
Computational trick (unphysical):
One interchanges the dependent and independent variable
this function is single valued (unique)
i.e. one chooses M and calculates
H afterwards

35. FSM key references A.R.Williams, V.L.Moruzzi, J.Kübler, K.Schwarz,
Bull.Am.Phys.Soc. 29, 278 (1984)
K.Schwarz, P.Mohn
J.Phys.F 14, L129 (1984)
P.H.Dederichs, S.Blügel, R.Zoller, H.Akai,
Phys. Rev, Lett. 53,2512 (1984)

36. Unusual magnetic systems
GMR (Giant Magneto Resistance)
half-metallic systems
e.g. CrO2
important for spintronics

37. ? Once upon a time, … Once upon a time, in the early 1980’s …
Peter Grünberg
“What happens if
I bring two ferromagnets close
–I mean really close–
together?” N S S N ?

38. Giant magnetoresistance (GMR)
Ferromagnet
Metal
Electrical
resistance:
RP <(>) RAP
The electrical resistance depends on
the relative magnetic alignment of the ferromagnetic layers
19% for
80% for RT
GMR is much larger than the anisotropic magnetoresistance (AMR)

39. 1988: … simultaneously, but independent …
“Does the electrical resistance depend on the magnetization alignment?”
Albert Fert
Peter Grünberg

40.
Scientific background

41. CrO2 half-metallic ferromagnet
CrO2 (rutile structure)
spin-up
spin-down
metallic
gap
important for spintronics

42. CrO2 DOS gap metallic The DOS features of CrO2 are qualitatively like
K.Schwarz,
CrO2 predicted as a
half-metallic ferromagnet,
J.Phys.F:Met.Phys. 16, L211 (1986)
The DOS features of CrO2 are qualitatively like
TiO2 (for spin-down)
RuO2 (for spin-up) 4 5 6 7 8 Ti V Cr Mn Fe Zr Nb Mo Tc Ru
spin
gap
metallic
all three compound crystallize in the rutile structure

43. Half-metallic ferromagnet
CrO2 (rutile structure)
spin-up
spin-down
metallic
gap

44. CrO2 spin-down (TiO2) spin-up (RuO2)

45. Magnetic structure of uranium dioxide UO2
R.Laskowski
G.K.H.Madsen
P.Blaha
K.Schwarz
topics
non-collinear magnetism
spin-orbit coupling
LDA+U (correlation of U-5f electrons)
Structure relaxations
electric field gradient (EFG) U O
R.Laskowski, G.K.H.Madsen, P.Blaha, K.Schwarz:
Magnetic structure and electric-field gradients of uranium dioxide: An ab initio study
Phys.Rev.B 69, (2004)

46. Atomic configuration of uranium (Z=92)
[Rn]
U [Xe] 4f d10 6s2 6p f3 6d1 7s2
core semi-core valence
Ej (Ryd)
nrel
j (relativ.)
n ℓ
ℓ-s
ℓ+s 7s
-0.25 6d
-0.29 5f
-0.17
-0.11 6p
-1.46
-2.10 6s
-3.40 5d
-7.48
-6.89 5p
-18.05
-14.06 5s
-22.57 4f
-27.58
-26.77
... 1s
delocalized
core-like

47. non-collinear magnetism in UO2
collinear 1k non-collinear 2k or 3k-structure

48. UO2 2k structure, LDA+SO+U
Magnetisation direction
perpenticular at
the two U sites (arrows)
Magnetisation density (color) U O

49. Magnetism with WIEN2k

50. Spin polarized calculations

51. Run spin-polarized, FSM or AFM calculations

52. Various magnetism cases

53. Thank you for your attention